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UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ Information Inequalities, variational inference and accelerated sensitivity screening

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst Turing Gateway to Mathematics - December 2015

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Outline:

◮ Uncertainty Quantification Information Inequalities:

◮ What do Information metrics and Variational Inference imply for

Quantities of Interest (observables)?

◮ Methods that scale appropriately for:

◮ High dimensional state/parameter space systems ◮ Long times in the case of dynamics

◮ Inequalities provide a tool to address UQ, inference and

transferability questions in CG.

◮ Demonstrate methods in accelerated sensitivity screening for

complex chemical reaction networks and Langevin molecular dynamics.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Acknowledgments

◮ Paul Dupuis (Brown University) ◮ Petr Plechac (University of Delaware) ◮ Luc Rey-Bellet (University of Massachusetts, Amherst) ◮ Dion Vlachos (Chem. Engineering & EFRC, University of Delaware) ◮ Georgios Arampatzis (ETH, Zurich) ◮ Yannis Pantazis (University of Massachusetts, Amherst) ◮ Kostis Gourgoulias (University of Massachusetts, Amherst) ◮ Jie Wang (University of Massachusetts, Amherst)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Why information-based methods?

◮ Pseudo-distance between probability measures P, Q:

RKL (P | Q) :=

• log

dP dQ

• dP

◮ Properties: (i) RKL (P | Q) ≥ 0 and

(ii) RKL (P | Q) = 0 iff P = Q a.e.

◮ Other probability metrics and divergences:

Total Variation, Hellinger, χ2, F-divergence, etc.

◮ Is relative entropy special? see later. ◮ Drawbacks: need absolute continuity, i.e. some probability models

cannot be compared. Need other methods (stochastic coupling, Malliavin calculus, etc)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Information metrics in inference, model selection and UQ

◮ Variational inference methods in machine learning ◮ Variational inference for building coarse-grained models in materials ◮ Information metrics for UQ and sensitivity analysis of stochastic

models

◮ Information metrics for quantifying predictive skill in model

selection/reduction

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Variational inference and coarse-graining

Loss of Information in Coarse-Graining, CG error, model fidelity : K., Vlachos J.

• Chem. Phys. (2003), Majda, Abramov (2006), K., Plechac, Rey-Bellet,

Tsagkarogiannis (2014), Chen, Tong, Majda (2014) ... RKL (P | Q) = N × O(ǫp) , N = system size, ǫ = tolerance CG Parametrizations via Variational Inference: Shell (2008, 2012), Noid et al (2011), Espanol, Zuninga (2011), Bilionis, Koutsourelakis (2012), Bilionis, Zabaras (2013), K., Plechac (2013) ... min

θ RKL (P | Q(θ)) .

Machine/Statistical Learning via Variational inference: Amari (1998), Jordan et al (1999), Bottou (2003), Wainright, Jordan (2008), Hoffman, Blei, Wang, Paisley (2013)...

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Coarse-graining in molecular simulations

◮ Build parametrized approx. of Gibbs states:

¯ µapp ∼ e−β ¯

Happ(η;θ) ◮ Find optimal values of parameters θ∗, such that

min

θ

• i

|Eµ[φi] − E¯

µapp[φi]|2

for selected observables φi, e.g. F. Muller-Plathe, Chem. Phys. Chem. (’02)

◮ Parametrization depends on specific observable(s) φ. ◮ Special case: Force-matching methods, [G. Voth and collaborators]

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Coarse Graining-Parameterization-Transferability

◮ Need to ensure accurate simulation of other observables, which are

not part of the parameterization, i.e.

◮ Can we improve the ”transferability”/predictability of the

parametrizations?

◮ Recall the CKP inequality: for any observable φ,

|EP[φ] − EQ[φ]| ≤ ||φ||∞

• 2RKL (P | Q)

RKL (P | Q): Loss of Information during Coarse Graining

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Sensitivity analysis - Information Theory Methods

◮ Consider parametrized probabilistic models Pθ(x)dx. ◮ Info Theory concepts: Relative entropy, Fisher Information matrix,

Mutual information, etc. RKL

• Pθ | Pθ+ǫ

= Loss of Information due to perturbation by ǫ

• H. Liu, W. Chen, and A. Sudjianto, J. Mech. Des. (2006).
• N. Ludtke et al., J. Royal Soc., Interface (2008).
• A. J. Majda and B. Gershgorin, Proc. Natl. Acad. Sci. (2010).
• M. Komorowski et al, Proc. Natl. Acad. Sci. (2011).

◮ The PDF is assumed known, e.g. Gibbs µ ∼ Z −1e−H(σ). Allows for

explicit calculations of R: RKL

• µθ | µθ+ǫ

∼ Eµθ[Hθ+ǫ − Hθ] + log Z θ+ǫ Z θ

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Sensitivity analysis - Information Theory Methods

◮ Consider parametrized probabilistic models Pθ(x)dx. ◮ Info Theory concepts: Relative entropy, Fisher Information matrix,

Mutual information, etc. RKL

• Pθ | Pθ+ǫ

= Loss of Information due to perturbation by ǫ

• H. Liu, W. Chen, and A. Sudjianto, J. Mech. Des. (2006).
• N. Ludtke et al., J. Royal Soc., Interface (2008).
• A. J. Majda and B. Gershgorin, Proc. Natl. Acad. Sci. (2010).
• M. Komorowski et al, Proc. Natl. Acad. Sci. (2011).

◮ The PDF is assumed known, e.g. Gibbs µ ∼ Z −1e−H(σ). Allows for

explicit calculations of R: RKL

• µθ | µθ+ǫ

∼ Eµθ[Hθ+ǫ − Hθ] + log Z θ+ǫ Z θ

◮ However, typically this is not the case in dynamics, non-equilibrium

systems, non-gaussian fluctuations, etc. Information metrics in path space. KL divergence per unit time.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Complex Reaction Networks

"

from)Ref.3)Right:)Example)of)reaction)network)of)a)small)oxygenate)(ethanol)on)platinum)catalyst).)

10 102 103 104 105 106 Intermediates Reactions

Number of Calculations

Sugar size Number of parameters Left: Number of parameters in metal-catalyzed upgrade of small biomass derivatives for the production of renewable fuels and chemicals

• vs. molecular size. The reaction network even of typical sugars, such as glucose, entails nearly a million of parameters. Right: Example of

reaction network of a small oxygenate (ethanol on platinum catalyst); the thickness of lines indicates the reaction flux.

• J. E. Sutton and D. G. Vlachos. Chem. Engin. Sci. 2015.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Deterministic Sensitivity Analysis - Background

◮ System of ODEs:

˙ y = f (y; θ) , y(0) = y0 ∈ RN

• Goal: Perform SA on the model parameters θ ∈ RK.

◮ Define sensitivity indices:

sk = ∂y ∂θk

◮ A new system of ODEs is derived and augmented to the previous:

˙ sk = ∂f ∂y sk + ∂f ∂θk , k = 1, ..., K

• need to solve K × N additional equations.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Challenges

◮ How ”predictive”? Information Inequalities and Observables/QoIs: ◮ For example, Csiszar-Kullback-Pinsker, (Observables/QoIs f )

|EP[f ] − EQ[f ]| ≤ ||f ||∞

• 2RKL (P | Q)

◮ Do such inequalities scale with system size N >> 1 ? ◮ Do the scale for long-time dynamics, i.e. a time window T >> 1? ◮ Applicable to non-equilibrium systems?

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Distance/Divergence between Probabilistic models

◮ The Total Variation Distance between P and Q :

TV (P, Q) = supA |

• A

(p − q)dµ |= 1 2

• | p − q | dµ;

◮ 0 ≤ TV (P, Q) ≤ 1

◮ The Kullback Divergence between P and Q :

R(P || Q) =

• log dP

dQ dP;

◮ The Hellinger distance between P and Q :

H(P, Q) = (

• (

√ dP −

• dQ)2)1/2 = (
• (√p − √q)2dµ)1/2;

◮ 0 ≤ H2(P, Q) ≤ 2

◮ The χ2 divergence between P and Q :

χ2(P || Q) =

• ( dP

dQ − 1)2dQ.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Inequalities

◮ Pinsker’s Inequality

TV (P, Q) ≤

• R(P || Q)/2

◮ Le Cam’s Inequality

TV (P, Q) ≤ H(P || Q)

◮ TV (P, Q) ≤ H(P, Q) ≤

• R(P || Q) ≤
• χ2(P || Q)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Example - Mean field models - Independence

◮ Product probability measures PN and QN where N ≫ 1

PN(σ1, ..., σN) =

N

• i=1

Pi(σi) , QN(σ1, ..., σN) =

N

• i=1

Qi(σi)

◮ Mean field class of probabilistic models ◮ Average observable on σ = (σ1, ..., σN):

f (σ) = 1 N

N

• i=1

g(σi)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ Bounds Using CKP

◮ Observable: Csiszar-Kullback-Pinsker inequality:

| EPN (f ) − EQN (f ) | ≤ f ∞

• 2R(PN || QN).

◮

R(PN || QN) = NR(P || Q)

◮

| EPN (f ) − EQN (f ) | ≤ √ Ng∞

• 2R(P || Q).

Conclusion

• 1. g∞ can easily take large values, e.g. reaction networks;
• 2. When N ≫ 1, the bound blows up.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ Bounds - Hellinger Distance

◮ Observable: | EPN [f ] − EQN [f ] |≤|| f ||∞ H(PN || QN) ◮

H2(PN || QN) = 2

• 1 − (1 − H2(P || Q)

2 )N

• ◮

| EPN (f ) − EQN (f ) | ≤ g∞

• 2
• 1 − (1 − H2(P || Q)

2 )N

• .
• 1. g∞ can easily take large values;
• 2. Consider P = Q, | EPN(f ) − EQN(f ) |≡| EP(g) − EQ(g) |, but
• 2
• 1 − (1 − H2(P || Q)

2 )N

• N→∞

→ √ 2.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ Bounds - Chapman Robbins Bound

◮ Chapman Robbins Bound (related to the Cramer Rao bound):

| EPN (f ) − EQN (f ) |≤

• VarPN (f )
• χ2(QN || PN)

◮

χ2(QN || PN) =

• 1 + χ2(Q || P)

N − 1

◮

| EPN (f ) − EQN (f ) | ≤

• 1

N VarP(g) (χ2(Q || P) + 1)N − 1 N VarP(g).

Consider P = Q, | EPN(f ) − EQN(f ) |≡| EP(g) − EQ(g) |, but 1 N VarP(g)

• χ2(Q || P) + 1

N → ∞ as N → ∞.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ Bounds - Goal-Oriented Divergence1

Ξ−(Q || P; f ) ≤ EP[f ] − EQ[f ] ≤ Ξ+(Q || P; f ). Ξ+(Q || P; f ) = infc>0{ 1 c

• Λp,f (c) + 1

c R(Q || P)} Ξ−(Q || P; f ) = supc>0{− 1 c

• Λp,f (c) − 1

c R(Q || P)}

◮ Cumulant generating function:

• ΛP,f (c) = logEP[ec(f −EP [f ])];

Properties:

◮ Ξ+(Q || P; f ) ≥ 0 and Ξ−(Q || P; f ) ≤ 0; ◮ Ξ±(Q || P; f ) = 0 if and only if Q = P or f is constant P-a.s.; ◮ Divergence contains information on observable f ; ◮ Linearization: Ξ±(Q || P; f ) ≈ ±

• VarP(f )
• 2R(Q || P)

1Dupuis, M. K., Pantazis, Plechac (also for path-space) SIAM/ASA J. of Uncert. Quant.’15; earlier work: Chowdhary and Dupuis,

ESAIM ’13; Li and Xiu, SISC ’12 Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ Bounds - Goal-Oriented Divergence

◮ Observable f (σ) = 1 N

N

i=1 g(σi).

Ξ−(QN || PN; f ) ≤ EPN[f ] − EQN[f ] ≤ Ξ+(QN || PN; f ).

◮ Ξ±(QN || PN; f) = Ξ±(Q || P; g)

e.g. consider the linearization: Ξ±(QN || PN; f ) ≈ ±

• VarPN(f )
• 2R(QN || PN)

= ±

• 1

N VarP(g)

• 2NR(Q || P)

≈ Ξ±(Q || P; g)

◮

| EPN[f ] − EQN[f ] |

• VarP(g)
• 2R(Q || P)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

◮ The first three bounds fail even in the independent case case with

N ≫ 1;

◮ The bound by Goal-Oriented Divergence works well in the mean field

case with high dimension N ≫ 1 (scale with the system size);

◮ Question: How does it work in the correlated case, for example

time-series observables with T ≫ 1 or Gibbs/Boltzmann distributions defined on a high-diemnsional space?

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Example:Gibbs/Bpltzmann Distributions

◮ Λ: lattice domain ◮ Hamiltonians: HΦ

Λ(σ) = X⊂Λ ΦX(σ), HΨ Λ(σ) = X⊂Λ ΨX(σ)

◮ ΦX(σ) and ΨX(σ) are interactions of spins in X; notation can include

multi-body correlations, etc. [D. Ruelle, B. Simon]

◮ Gibbs/Boltzmann distribution:

µΦ(σ) = 1 ZΦ e−βHΦ

Λ(σ)

µΨ(σ) = 1 ZΨ e−βHΨ

Λ(σ)

◮ Example of observable: ”magnetization” per unit volume

f = 1 | Λ |

• i

σi

◮ Other observables: susceptibility, spatial correlations, power spectral density,

energy, etc.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ bounds for Gibbs/Boltzmann

◮ ||| Φ |||=

X∋0 | X |−1|| ΦX ||∞; [D. Ruelle; B. Simon]

1 | Λ |R(µΨ || µΦ) = 1 | Λ |(logZΦ − logZΨ) + β 1 | Λ |EµΨ[HΦ

Λ(σ) − HΨ Λ(σ)]

≤ ||| Φ − Ψ ||| +β ||| Φ − Ψ |||

◮

| Λ | VarµΦ(f ) = 1 | Λ |

• i,j

EµΦ[(σi − EµΦ[σi])(σj − EµΦ[σj])] = O(1)

(away from critical points)

◮ UQ bounds for N ≫ 1. For what observables f ?

| EµΨ[f ] − EµΦ[f ] |

• | Λ | VarµΦ(f )
• 2

| Λ |R(µΨ || µΦ) ∼ ||| Φ − Ψ |||1/2

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

||| Φ − Ψ |||

◮ Ising model ◮ σi ∈ {−1, 1}, i ∈ {..., −1, 0, 1, 2, ...}

HΦ

Λ(σ) =

• X∈Λ

ΦX(σ) = −

• |i−j|=1

Jijσiσj −

• i

hiσi HΨ

Λ(σ) =

• X∈Λ

ΨX(σ) = −

• |i−j|=1

˜ Jijσiσj −

• i

˜ hiσi

◮ Bounds independent of N:

||| Φ − Ψ ||| =

• X∋0

| X |−1|| ΦX − ΨX ||∞ = 1 2 (| J01 − ˜ J01 | + | J−10 − ˜ J−10 |)+ | h0 − ˜ h0 |

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Outline

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ for dynamics

Compare probabilistic models P = P[0,T], Q = Q[0,T] on path (time series) space: Ξ−(P, Q; f ) ≤ EQ[f ] − EP[f ] ≤ Ξ+(P, Q; f ) ,

◮ Example: Discrete-time Markov processes {Xt}T t=0; ◮ Ergodic averages of observables (T ≫ 1) - depend on the time

series: F(x0, . . . , xT−1) = 1 T

T−1

• i=0

f (xi), 1 T

T−1

• i=0

f (xi, xi+l), ...

• 1P. Dupuis, M. K., Y. Pantazis, P. Plechac SIAM/ASA J. of UQ 2015

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

◮

1 T R(Q[0,T] || P[0,T]) = H (Q | P) + 1 T R(ν || µ) where the Relative Entropy Rate is H (Q | P) = Eµ

• q(σ, σ′) log q(σ, σ′)

p(σ, σ′) d σ′

• ◮ Integrated Autocorrelation Time (IAT):

TVarP[0,T](F) = Varµ(f ) + 2

T

• k=1

(1 − k T )Af (k) → Varµ(f ) + 2

∞

• k=1

Af (k) := τ(f ) where Af (t) = EP[0,T][(f (X0) − Eµ[f (X0)])(f (Xt) − Eµ[f (X0)])];

◮

| EP[0,T][F] − EQ[0,T][F] |

• TVarP[0,T](F)
• 2

T R(Q[0,T] || P[0,T]) =

• TVarP[0,T](F)
• 2(H (Q | P) + 1

T R(µ || µ)) ∼ τ(f )1/2H (Q | P)1/2

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Outline

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Sensitivity Analysis - Relative Entropy Rate (RER)

RKL

• Qθ

0,T | Qθ+ǫ 0,T

• = Loss of Information (in time-series) due to perturbation by ǫ

◮ For long times T >> 1, RER is viewed as measure of parameter

sensitivity: RKL

• Qθ

0,T | Qθ+ǫ 0,T

• = TH
• Qθ

0,T | Qθ+ǫ 0,T

• + RKL
• µθ | µθ+ǫ

◮ θ: parameter vector (for local sensitivity it is fixed) ◮ ǫ: parameter vector perturbation

◮ (RER): H

• Qθ

0,T | Qθ+ǫ 0,T

• = Eµθ
• pθ(σ, σ′) log

pθ(σ,σ′) pθ+ǫ(σ,σ′)d σ′

◮ p(σ, σ′): transition probabilities (local dynamics). ◮ Eµθ.... steady-state sampling: ¯

H(n)

2

= 1

n

n−1

i=0 log pθ(σi ,σi+1) pθ+ǫ(σi ,σi+1)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Path-space Fisher Information Matrix (FIM)

Under a smoothness assumption on θ, (checkable, on the rates only!) H

• Qθ

0,M | Qθ+ǫ 0,M

• = 1

2ǫTFH

• Qθ

0,M

• ǫ + O(|ǫ|3)

The path Fisher Information Matrix is defined as FH

• Qθ

0,M

• = Eµθ
• E

pθ(σ, σ′)∇θ log pθ(σ, σ′)∇θ log pθ(σ, σ′)Td σ′

• ◮ Spectral analysis of FIM gives the most/least sensitive directions.

◮ Sparse structure of the path FIM-see examples below. ◮ (Stochastic) derivative-free sensitivity analysis method. ◮ Characterizes robustness on simultaneous parameter perturbations. ◮ Determines parameter identifiability, e.g. Cramer-Rao-type Thms

1Pantazis, K., J. Chem. Phys. (2013); K., Y. Pantazis, D. Vlachos, BMC Bioinformatics, (2013), Arampatzis, K. Pantazis PLOS 1

(2015). Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Path-space FIM - Sparsity

◮ For reaction networks, Langevin, etc. we typically have a block diagonal

structure in the FIM

◮ Scalable computations - FIM scales linearly in the number of parameters. ◮ Contains key information:

◮ Graph structure ◮ Dynamics on the graph: reaction rates and their functional form

Reactions Parameters Fisher Information Matrix

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Sensitivity for Langevin dynamics

◮ Potential: V (q) = Vangle(q) + Vbond(q) + VLJ(q) ◮ Bond potential: Vbond(rij) = 1

2 Kb(r0 − rij)2

◮ Angle potential: Vangle(θijk) = 1

2 Kθ(θ0 − θijk)2

◮ Interaction potential:

VLJ(rij) =    4ǫLJ

• σLJ

rij

12 −

• σLJ

rij

6 if rij < rcut.

• therwise.

◮ Parameter vector:

θ = [ǫC−C

LJ

, σC−C

LJ

, ǫC−H

LJ

, σC−H

LJ

, ǫH−H

LJ

, σH−H

LJ

, Kb, r0, Kθ, rθ]T

• Relative Entropy Rate: (related to force-matching in coarse-graining)

H(Qθ|Qθ+ǫ) = 1 2 Eµθ[(F θ+ǫ(q) − F θ(q))T (σσT )−1(F θ+ǫ(q) − F θ(q))]

• Fisher Information Matrix:

FH(Qθ) = Eµθ[∇θF θ(q)T (σσT )−1∇θF θ(q)] ,

• 1V. Harmandaris, A. Tsourtis, M.K., Pantazis J. Chem. Phys. (2015)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Outline

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

UQ Bounds = ⇒ Sensitivity Screening

◮ UQ bounds:

|EQθ

0,T [f ] − EQθ+ǫ 0,T [f ]| ≤

• Varθ(f )
• 2RKL
• Qθ

0,T | Qθ+ǫ 0,T

• + O(|ǫ|2)

◮ Sensitivity indices of observable f :

Sf (θk) =

∂ ∂θk Eµθ[f ]

◮ Direct approaches: Finite Diff. with coupling, linear response,... ◮ An implementable bound can be derived using the path-space SA approach:

|Sf (θk)| ≤

• ∞
• k=−∞

Af (k)

• FH(Qθ)k,k

◮ where Af (k) =< f (x0), f (xk) >µθ is the stationary correlations. ◮ ∞

k=−∞ Af (k) Integrated Autocorrelation Time (IAT) - anyhow necessary

to estimate variance.

◮ FH(Qθ): path FIM.

◮ Interpretation: FIM-insensitive parameters are also f -insensitive.

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Screening Strategy - Information Metrics & Observables

◮

Biological network describing Epidermal Growth Factor Receptor. [Kholodenko et.al., J. Biol. Chem., 1999] Data from: http://www.ebi.ac.uk/biomodels-main/BIOMD0000000048

◮

47 reactions, 23 species, 23 observables, 50 parameters, 23 × 50 = 1150 sensitivities

◮

If we include species correlations we have 12,650 sensitivities, etc.

5 10 15 20 25 30 35 40 45 10 20 30 40 50 60 Sensitivity Index for EGFR model. #Sensitivities=1150 cummulative sqrt(RER) cummulative sqrt(IAT) +10% (650) +15% (224) +25% (165) 50% (111)

Screening model sensitivities based on the path-FIM upper bound: |Sf (θk)| ≤

• ∞
• i=−∞

Af (i)

• FH(Qθ)k,k

Arampatzis, Pantazis, Katsoulakis PLOS 1, (2015) Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Example - Unsorted parameters/observables

Figure: Unsorted SA: (a) Sensitivity Analysis in time interval [0, 50], (b) Sensitivity Analysis in time interval [50, 100] (steady state)

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Path-space FIM - Sparsity

◮ For reaction networks, Langevin, etc. we typically have a block diagonal

structure in the FIM

◮ Scalable computations - FIM scales linearly in the number of parameters. ◮ Contains key information:

◮ Graph structure ◮ Dynamics on the graph: reaction rates and their functional form

Reactions Parameters Fisher Information Matrix

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Stochastic SA - Observable-based

◮ Finite differencing:

S(θ, t) ≈ EPθ+ǫ

t

[f (x)] − EPθ

t [f (x)]

ǫ + O(ǫ)

◮ Variance of the estimator controlled by:

var (f (xθ+ǫ

t

) − f (xθ

t )) = var (f (xθ+ǫ t

)) + var (f (xθ

t ))

− 2cov (f (xθ+ǫ

t

), f (xθ

t ))

High Variance due to sampling of two different stochastic processes. Can we ”couple” the two processes?

2 4 6 8 10 12 14 16 18 20 0.2 0.4 0.6 0.8 time coverage Coupled processes unperturbed perturbed 2 4 6 8 10 12 14 16 18 20 0.2 0.4 0.6 0.8 time coverage Uncoupled processes unperturbed perturbed

Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

Stochastic Coupling Methods

Simulate joint process (xθ+ǫ

t

, xθ

t ) with constrained marginals: ◮ Common Random Number/Common

Reaction Path approach: [Rathinam, Sheppard, Khammash, J.

• Chem. Phys., 2010]

◮ Markov (xθ+ǫ

t

, xθ

t ) for well–mixed

systems: [Anderson, SIAM Numerical Analysis, 2012], [Srivastava, Anderson, Rawlings, J. Chem. Phys., 2013]

◮ Markov (xθ+ǫ

t

, xθ

t ) for spatial KMC -

couplings based on coupling optimization principle: [Arampatzis, K. , J.Chem.Phys., 2014]

◮ Efficient methods, but impractical for

systems with a large # of parameters

5 10 15 20 25 30 35 40 10

1

10

2

10

3

10

4

time variance Uncoupled CRN Trivial Coupling Unoptimized Coupling Optimized Coupling Variance for different coupling methods in Lattice KMC: [Arampatzis, K., J.Chem.Phys., ’14] Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit

UQ bounds for dynamics Accelerated Sensitivity Screening UQ bounds and sensitivity screening

References

◮

Sensitivity Analysis in Path Space (information-theoretic, goal-oriented) A Relative Entropy Rate Method for Path Space Sensitivity Analysis of Stationary Complex Stochastic Dynamics, Y. Pantazis, M.K., J. Chem. Phys. (2013). Parametric Sensitivity Analysis for Biochemical Reaction Networks based on Pathwise Information Theory, M. K., D. Vlachos, Y. Pantazis, BMC Bioinformatics, (2013). Measuring the irreversibility of numerical schemes for reversible stochastic differential equations, M. K. Y. Pantazis, L. Rey-Bellet., ESAIM: Math. Model. Num. Analysis, (2014). Goal-oriented sensitivity analysis for lattice kinetic Monte Carlo simulations, M. K. and G. Arampatzis, J. Chem. Phys., (2014). Accelerated Sensitivity Analysis for High-Dimensional Stochastic Reaction Networks, G. Arampatzis, Y. Pantazis, M. K. PLOS1 (2015). Sensitivity Bounds and Error Estimates for Stochastic Models, P. Dupuis, M. K., P. Plechac, Y. Pantazis SIAM UQ (under revision) (2015). Parametric Sensitivity Analysis for Stochastic Molecular Systems using Pathwise Information Metrics, A. Tsourtis, V. Harmandaris, M. K., Y.Pantazis J. Chem. Phys. (2015).

◮

Coarse-graining (path-space, multi-body effects, non-equilibrium) Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems, M. K., P. Plechac,

• J. Chem. Phys. (2013).

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions, M. K., P. Plechac, L. Rey-Bellet and D. Tsagkarogiannis, Math. Comp., (2014). Spatial two-level interacting particle simulations and information theory-based error quantification, E. Kalligiannaki, M. K. , P. Plechac SIAM Sci. Comp., 36, A634A667 (2014). Measuring the irreversibility of numerical schemes for reversible stochastic differential equations, M. K. Y. Pantazis, L. Rey-Bellet., ESAIM: Math. Model. Num. Analysis, (2014). Path-space variational inference for non-equilibrium coarse-grained systems, V. Harmandaris, E. Kalligiannaki, M. K. and P. Plechac, J. Comp. Phys., submitted, (2015). See also: Markos Katsoulakis’ Homepage, ResearchGate Profile Markos Katsoulakis Mathematics & Statistics University of Massachusetts, Amherst UQ Information Inequalities, variational inference and accelerated sensitivit